Two players start with a pile of stones.
1st player takes any number of stones from the pile, but not all of them.
Then the players alternate, taking up to twice as many stones
as their opponent just took.
The player taking the last stone (or stones) is the winner.

Question: If you and your friend start with 100 stones, what is the smallest amount of stones you should take on your 1st turn, and how will you proceed to win the game?

Here's a table showing the number of stones you should take on each turn to force a win, depending on how many stones are remaining at the beginning of your turn - e.g. when there are 100 stones remaining, you can take 3 and eventually force a win.  These aren't necessarily the only valid moves (as I noted in my previous post, you can also start with 11 and force a win) but following this table will lead to a victory. 

Remaining Take
100 3
99 2
98 1
97 *
96 2
95 1
94 5
93 1
92 3
91 2
90 1
89 *
88 1
87 3
86 2
85 1
84 8
83 2
82 1
81 5
80 1
79 3
78 2
77 1
76 21
75 2
74 1
73 5
72 1
71 3
70 2
69 1
68 13
67 1
66 3
65 2
64 1
63 8
62 2
61 1
60 5
59 1
58 3
57 2
56 1
55 55
54 2
53 1
52 5
51 1
50 3
49 2
48 1
47 13
46 1
45 3
44 2
43 1
42 8
41 2
40 1
39 5
38 1
37 3
36 2
35 1
34 34
33 1
32 3
31 2
30 1
29 8
28 2
27 1
26 5
25 1
24 3
23 2
22 1
21 21
20 2
19 1
18 5
17 1
16 3
15 2
14 1
13 13
12 1
11 3
10 2
9 1
8 8
7 2
6 1
5 5
4 1
3 3
2 2
1 1

Edited on June 30, 2014, 3:25 pm

Posted by tomarken on 2014-06-30 13:48:44